Av(12345, 12354, 12435, 12453, 12534, 12543, 13452, 13542, 23451, 23541)
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Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+9 x^{2}-x \right) \sqrt{x^{2}-6 x +1}-2 x^{4}+15 x^{3}-18 x^{2}-9 x +2}{2 x^{2}-12 x +2}\)
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Counting Sequence
1, 1, 2, 6, 24, 110, 540, 2758, 14448, 77022, 415860, 2267078, 12452616, 68814798, 382168332, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-6 x +1\right) F \left(x \right)^{2}+\left(x -2\right) \left(2 x +1\right) \left(x^{2}-6 x +1\right) F \! \left(x \right)-2 x^{4}+3 x^{3}-17 x^{2}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - \frac{2 \left(n - 2\right) a{\left(n \right)}}{n + 3} + \frac{3 \left(5 n + 8\right) a{\left(n + 3 \right)}}{n + 3} + \frac{3 \left(7 n - 6\right) a{\left(n + 1 \right)}}{n + 3} - \frac{3 \left(19 n + 9\right) a{\left(n + 2 \right)}}{n + 3}, \quad n \geq 5\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 8 rules.

Finding the specification took 2 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{6}\! \left(x , y\right) &= x F_{6}\! \left(x , y\right) y +F_{6}\! \left(x , y\right)^{2}-2 F_{6}\! \left(x , y\right)+2\\ F_{7}\! \left(x \right) &= x\\ \end{align*}\)